Abstract

In this paper we use the recently suggested conjecture about the integral representation for the flat coordinates on Frobenius manifolds, connected with the isolated singularities, to compute the flat coordinates and Saito primitive form on the space of the deformations of Gepner chiral ring ˆSU(3)4.

We verify this conjecture comparing the expressions for the flat coordinates obtained from the conjecture with the one found by direct computation. The considered case is of a particular interest since together with the relevant and marginal deformations it has one irrelevant deformation.

1 Introduction

This paper is a continuation of the previous series of works [1, 2, 3] and dedicated to a new approach to computations of the flat coordinates of the Frobenius manifolds connected
with isolated singularity [5, 4].

Frobenius structure arises in the three kinds of models of QFT and string theory. Namely, in the models of two-dimensional topological CFT [6], in the models of space-time supersymmetric compactifications of string theory on Calabi-Yau manifolds [7, 8, 9, 10] and in the models of Polyakov noncritical string theory [11].

One of the key ingredients for the solution of this models is knowledge [6, 12, 13] of the flat coordinates on the corresponding Frobenius manifolds.

A new method for the computation of the flat coordinates, based on the conjecture about an integral representation, has been suggested for the models connected with the simple ADE singularities in [1]. In [2, 3] this method and the conjecture itself were formulated for the case of the general isolated singularity. Also, in these works the conjecture was verified for the model with a number of relevant and one marginal deformations.

The aim of this work is to use and verify the conjecture in the computation of the flat coordinates on Frobenius manifolds of deformations the Gepner chiral ring ˆSU(3)4.
This model has one marginal and one irrelevant deformation. We verify the results obtained by the use of the conjecture by comparing with the results of the direct computation.

In section 2 we briefly review Dubrovin-Saito theory. In section 3 we formulate the conjecture of the integral representation for flat coordinates in case when Saito primitive form is not trivial what happens when the marginal and irrelevant deformations take place. In section 4 we review some needed facts about the deformation of the Gepner chiral ring ˆSU(3)4 for which our computations are performed. In section 5 we explain how one can find metrics and flat coordinates by a direct way from the Frobenius manifold structure axioms, compare the results of both computations and find their coincidence. In section 6 we discuss moduli of the primitive form and their connection with the resonances in the ring. We provide more detailed expressions for the flat coordinates and the primitive form in the appendix.

2 Preliminaries

In this section, we review the role of the flat coordinates of the Frobenius manifold for the case of topological CFT which comes from Witten twist and restriction of space of states on the chiral sector of the N=2 SCTF Landau-Ginzburg model.

In these models superpotential W0[Φ1,…,Φn] depends on n fundamental chiral fields. These fields generated chiral ring R0 and we will denote basis in it as Φα for α=1,…,M. Here M=dimR0, first fields with α=1,…,n will label generators of the ring and Φ1=1 is a unity operator.

The chiral ring R0 is isomorphic to the ring of the polynomials of xi

R0=Cn[x1,…,xn]/{∂W0∂xi},

(2.1)

where {∂W0∂xi} denotes the ideal generated by the partial derivatives of the polynomial W0[xi].

It was shown in [6] that for computation of the correlation functions of the fields Φα and its superpartners Φ(1,1)α=G−−1/2G+−1/2Φα it is needed and sufficient to know the two-point functions

ηαβ=⟨ΦαΦβ⟩,

(2.2)

together with the perturbed three-point function

Cαβγ(s1,…,sM)def=⟨ΦαΦβΦγexp(M∑λ=1sλ∫Φ(1,1)λd2z)⟩.

(2.3)

It was also shown in [6] that ηαβ is non-degenerate and s-independent and Cαβγ(s) can be expressed through a prepotential (or free eneregy) F

These relations together with the evident property Cγαβ=Cγβα mean that Cγαβ(s) are structure constants for a commutative, associative algebra or a ring R with unity which depends on the parameters {sα}.

The properties of ηαβ and Cγαβ(s) mean indeed that we have the Frobenius manifold structure [5] and sμ are nothing but the flat coordinates on this manifold, i.e. such coordinates in which the Riemann metric ηαβ is constant.

The crucial fact, found in [6],
which makes possible to exactly solve the topological models of such kind, is that the Frobenius manifold, defined by Cαβγ(s) and ηαβ, coincides with a Frobenis manifold defined by the versal deformations W(x,t) of a superpotential W0

W(x,t)def=W0(x)+M∑α=1tαeα(x).

(2.6)

Here {eα} is a basis of the ring R0 (2.1), e1(x)=1 is a unity element of R0.
The corresponding ring, defined by W(x,t) as the ring of polynomials of xi,

RW=Cn[x1,…,xn]/{∂W∂xi}.

(2.7)

The structure constants ˜Cγαβ(t) of RW in the basis {eα} are defined by the relations

eαeβ=˜Cγαβ(t)eγmod{∂W∂xi}.

(2.8)

The Riemann metric gαβ(t) is defined as the Grotendick residue in the terms of the Saito primitive form [4]

Ω(x,t)=λ(x,t)dx1∧⋯∧xn

(2.9)

as follows

gμν=Resx=∞eμeνΩ∏i∂W/∂xi.

(2.10)

It was proved in [14] the primitive form does exist. Namely, there exist such a differential form Ω(x,t), that the structure constants ˜Cγαβ(t) (2.8) and Riemann metric gαβ(t) (2.10) satisfy the Dubrovin Frobenius manifold axioms:

˜Cραβ˜Cμργ

=˜Cραγ˜Cμρβ,

(2.11)

Rμνλσ[gαβ]

=0,

(2.12)

∇σ˜Cμνλ

=∇μ˜Cσνλ,

(2.13)

˜Cμνλ

=˜Cνμλ=˜Cμλν.

(2.14)

The deformation parameters {tα} are some coordinates on the Frobenius manifold.
The coupling constants sμ are the flat coordinates on it. They are functions of the deformation parameters {tα}.

The knowledge of these functions permit to express the perturbed three-point functions
Cαβγ and the prepotential F in terms of gμν
and ˜Cγαβ(t) .
Thus the determination of the functions sμ(t) and the primitive form Ω(x,t) is the major part of the solution of the topological Landau-Ginzburg model.

3 The flat coordinates through the oscillating integrals.

We will assume that the superpotential W0(x) is a quasihomogeneous polynomial associated to an isolated singularity

W0(Λρixi)=ΛdW0(xi),

(3.1)

where integer weights d=[W0] and ρi=[xi].

In this case we can choose the basis eα of the ring RW to be quasihomogeneous . We will denote its weights as degeα. The elements of the basis are called relevant, marginal or irrelevant if their weights satisfy correspondingly the relations degeα<d, degeα=d or degeα>d.

It was conjectured in [1, 3] that the flat coordinates are given by the following integral expression

sμ(t)=∑mα∈Σμ(∫γμexp(W0(x))∏αemααΩ)∏αtmααmα!,

(3.2)

where Σμ is specified by requirement for l.h.s. and r.h.s. of this equation to have the same weights. We will give the explicit expression for it below.

The cycles γμ form basis
for the homology Hn(Cn,ReW0=−∞) which defined

as limL→+∞Hn(Cn/{ReW0≤−L}).333In this limit, a homology class can be repesented by non-compact closed n-dimensional submanifold γ in Cn such that Re(W0) tends to −∞ at infinity, and therefore∫γeW0dx converges.

A simple example of the explicit choice of such cycles for n=1 has been given in [1] on
Figure 2. It illustrates this notion.

We fix the normalization of the coordinates by the requirement for the first term of the decomposition to be sμ=tμ+….

For computing the integrals in (3.2) we use the same way as in [3]. The main point of the computation is the following property of the oscillating integrals

∫γexp(W0(x))P1(x)dx=∫γexp(W0(x))P2(x)dx,

(3.3)

if there exist an (n−1)-form U such that

P1(x)dx=P2(x)dx+DW0U,

(3.4)

where DW0 is Saito differential

DW0=d+dW0∧.

(3.5)

The differential DW0 defines the Saito cohomology Hn on the space of n-forms.

The forms eμdx for μ=1,…,M can be chosen as a convenient basis in Hn.

Let us define a pairing between the elements of the basis eμdx in Hn and
the cycles γμ as

rμ,ν=∫γμexp(W0)eνdx.

(3.6)

We can choose the homology basis γμ to be dual to the cohomology basis eμdx.

The simplest choice of the dual basis is tempting to be

rμ,ν?=δμ,ν.

(3.7)

However, a more general possibility have to be considered. The reason for this is the occurrence of resonances. We will call resonance the case when the weights of some coordinates satisfy [sμ]−[sν]=0modd and [sμ]≠[sν]. We use the following choice
rμ,μ=1 for all μ and rμ,ν=0 if coordinates sμ and sν are not in resonance.444In the case of our interest ˆSU(3)4 this gives four parameters r1,14,r2,15,r14,1,r15,2. The last two are fixed by the normalization condition and the equation (3.14). However, r1,14,r2,15 stay to be free parameters. Doing in this way we get the expressions for the flat coordinates which depend on some extra parameters as it is predicted in [16].

Since eμdx form a basis of Hn, any n-form can be decomposed in it. In particular,

∏αekααdx=∑μBμ(k)eμdx+DW0U.

(3.12)

From the homogeneity requirements only such elements eμdx of the basis appear in the r.h.s of this equation whose weights are equal to those of the l.h.s module d. In case of the resonance a few elements of the same weights can appear in r.h.s. of (3.12). Their appearence in the oscilating integrals (3.6) is the reason of arising of the parameters rμ,ν in the expressions for sμ when eμ and eν are in a resonace.

In the case of our interest ˆSU(3)4, which is considered below, there are two resonances [s1]−[s14]=7 and [s2]−[s15]=7. We find that in this case two parameters r1,14 and r2,15, if they are not assumed to be equal zero, arise in the expressions for the flat coordinates derived from (3.2).

For given kα we can solve the equation (3.12) and find the coefficients Bμ(k) .
Substitution them into (3.10) gives

sμ(t)=∑mα∈Σμ,nα,lα∈ωA(n,l)Bμ(m+n)∏αtmα+lααmα!.

(3.13)

This formula gives expressions for sμ that depend on the unknown parameters A(n,l) of the primitive form.
We find these parameters using the normalisation conditions together with the equation [3]

∂sμ∂t1=δμ,1.

(3.14)

In such a way we arrive to the explicit expression for flat coordinates.
The final answer for the flat coordinates contains no free parameters besides those of rμ,ν, which correspond to the resonances.

4 The deformed chiral ring ˆSU(3)4

In the topological CFT,which is connected with the deformed chiral ring ˆSU(3)4[19],
the superpotential is

The first 13 elements of this basis are related to relevant deformations. The elements e14 and e15 are related to the marginal and irrelevant deformations correspondingly.

We computed the flat coordinates up to the 6th order in t by using the technique of the previous section.
The expressions for them up to the 2nd order are presented in the appendix.
We also put there the answer for the primitive form Ω up to the 2nd order in t.

5 Direct computation of flat coordinates

The expression for the flat coordinates (3.13) is a conjecture. This conjecture was tested in [2, 3] for the topological CFT connected with chiral ring ˆSU(3)3, where one marginal deformation takes place.

One of the main aims of this work is to check this conjecture for the case when there are also irrelevant deformations like it takes place for the model connected with the deformed Gepner chiral ring ˆSU(3)4 .

In order to do it, we have to compute the flat coordinates by the direct way. We did it perturbatively in overall t up to 4th order in t. The final answers are too lengthy to be presented here. Therefore, we will only outline the main steps of the calculation giving as many details as possible.

The metric on the Frobenius manifold is defined as

gμν=Resx=∞eμeνΩ∏i∂W/∂xi.

(5.1)

Instead of computing this residue we will follow the way used in [15]. Namely we rewrite the metric as

gμν=Cλμν(t)Resx=∞eλΩ∏i∂W/∂xi=Cλμν(t)hλ(t),

(5.2)

where hμ(t) are some unknown functions of t. These functions can be found from the Frobenius axioms

Rμνλσ[gαβ]

=0,

(5.3)

∇σCμνλ

=∇μCσνλ,

(5.4)

Cμνλ

=Cνμλ=Cμλν,

(5.5)

where Rμνλσ -Riemann curvature, Cμνλ is structure constants with index lowered by gαβ.

Using computer we found expression for the metric up to the 4th order in t. From equation (5.3) we found expressions for hμ(t) which still contain two parameters. After these equations (5.5) automatically satisfied. The solution of (5.4) fixes the value of one of the two parameters leaving only one. The fact that solution of equations (5.3-5.5) have one parameter is in perfect agreement with [16].

Finally, one can find flat coordinates from the equation

∂2sμ∂tα∂tβ=Γγαβ∂sμ∂tγ.

(5.6)

Since the metric found from equations (5.3-5.5) contains one parameter the flat coordinates will also contain a parameter. These results are in perfect agreement up to the fourth order with the one of section 3 if we impose the constraint r1,14=r2,15 .

6 Resonances and modules of primitive form

In this section we want to connect our results with one of [16]. At first we need to introduce some notation (in this section we will change several conventions in order for the formulas to coincide with [17]). We will label weights of basis diagrams as σi=degei/degW0. Basis in the ring must be ordered in such a way that σ1≤σ2≤⋯≤σM. It was proved in [16] that dimension D of the moduli space of the primitive form (or number of free parameters) is given by the formula

D=#{(i,j)|p(i,j)∈Z>0,i+j<M+1}+#{(i,j)|p(i,j)∈Zodd>0,i+j=M+1},

(6.1)

where p(i,j)=σi−σj, M is a dimension of the chiral ring, Z>0 are positive integers and Zodd>0 are odd positive integers.

Note that p(i,j) being integer is exactly the condition of resonance ([sμ]−[sν]=0modd and [sμ]≠[sν]) we used in our paper. Consider a few simple examples.

Thus the number of the resonances is five meanwhile the dimension of the moduli space is three.

The extra resonances are the origin of extra parameters rμ,ν in the expressions for the flat coordinates obtained from (3.2).

7 Conclusion

The comparison of two computations, performed in this work, shows that after imposing the constraint on the parameters r1,14=r2,15 the both expressions for the flat coordinates coincide.

This coincidence confirms the correctness of the Conjecture (3.2), now in the case when there are one marginal and one irrelevant deformations in the addition to the relevant ones.

In the same time, this comparison leads to the interesting question about the nature of the constraints which have to be imposed on the extra parameters rμ,ν which are predicted in [16].

Recently a new perturbative method to compute the flat coordinates and the primitive form
has been suggested in the papers [17, 18]. It would be interesting to understand the connection between this method and our approach.

Probably using the results of [17, 18] can help to prove the Conjecture (3.2) about the representation for flat coordinates through the oscillating integrals.

Acknowledgements. We thank A. Givental for the useful discussions and very valuable explanations on the homology Hn(Cn,ReW0=−∞).

The work was performed with the financial support of the Russian Science Foundation (Grant No.14-12-01383).

Appendix

Expressions for sμ up to second order in t

Here we present expressions for the flat coordinates obeyed via the conjecture. In order for this answers to coincide with the direct computation of section 5 one must set r1,14=r2,15.

Expressions for s1 and s2 are too lengthy and we don’t present them here.

Primitive form up to second order in t

We present the expression for the primitive form up to the second order in overall t. Note that decomposition (3.8) is overdetermined since any polynomial of eα can be reexpressed as polynomial of only e2 and e3. We used this freedom to express the primitive form linearly in eα.